5.3 Optimization Problems Involving Exponential Functions
Chapter
Chapter 5
Section
5.3
calcvecnelsonCh5.3lectureHQ 2 Videos
Solutions 24 Videos

Use graphing technology to graph each of the following functions. From the graph, find the absolute maximum and absolute minimum values of the given functions on the indicated intervals.

f(x) =e^{-x}-e^{-3x} on 0\leq x \leq 10

0.20mins
Q1a

Use graphing technology to graph each of the following functions. From the graph, find the absolute maximum and absolute minimum values of the given functions on the indicated intervals.

m(x) =(x + 2)e^{-2x} on x\in [-4, 4]

0.26mins
Q1b

Use the algorithm for finding extreme values to determine the absolute maximum and minimum values of the functions

f(x) =e^{-x}-e^{-3x} on 0\leq x \leq 10

1.40mins
Q2a

Determine the absolute max for m(x).

m(x) =(x + 2)e^{-2x} on x\in [-4, 4]

1.34mins
Q2b

The squirrel population in a small self-contained forest was studied by a biologist. The biologist found that the squirrel population, P, measured in hundreds, is a function of time, t, where t is measured in weeks. The function is P(t) = \frac{20}{1 + 3e^{-0.02t}}.

a. Determine the population at the start of the study, when t = 0.

b. The largest population the forest can sustain is represented mathematically by the limit as t \to \infty. Determine this limit.

1.28mins
Q3ab

The squirrel population in a small self-contained forest was studied by a biologist. The biologist found that the squirrel population, P, measured in hundreds, is a function of time, t, where t is measured in weeks. The function is P(t) = \frac{20}{1 + 3e^{-0.02t}}.

a. Determine the point of inflection.

b. Graph the function.

c. Explain the meaning of the point of inflection in terms of surreal population growth.

7.17mins
Q3cde

The net monthly profit, in dollars, from the sale of a certain item is given by the formula P(x) = 10^6[1 +(x -1)e^{-0.001x}], where x is the number of item sold.

Determine the number of items that yield the maximum profit. At full capacity the factory can produce 2000 items per month.

2.44mins
Q4a

The net monthly profit, in dollars, from the sale of a certain item is given by the formula P(x) = 10^6[1 +(x -1)e^{-0.001x}], where x is the number of item sold.

Determine the number of items that yield the maximum profit. At full capacity the factory can produce 500 items per month.

0.59mins
Q4b

Suppose that the monthly revenue in thousands of dollars, for the sale of x hundred units of an electronic item is given by the function R(x) = 40x^2e^{-0.4x} + 30, where the maximum capacity of the plant is 800 units. Determine the number of units to produce in order to maximize revenue.

3.08mins
Q5

A rumour spreads through a population in such a way that t hours after the rumour starts, the percent of people involved in passing it on is given by P(t) = 100(e^{-t} - e^{-4t}). What is the biggest percent of people involved in spreading the rumour within the first 3 h? When does this occur?

2.02mins
Q6

Small countries trying to develop an industrial economy rapidly often try to achieve their objectives by importing foreign capital and technology. Statistics Canada data show that when Canada attempted this strategy from 1867 to 1967, the amount of U.S. investment in Canada increased from about represented by the simple mathematical model C(t) = 0.015 \times 10^9 e^{0.07533t}, where t represents the number of years (starting with 1967 as zero) and C represents the total capital investment from US sources in dollars.

• Graph the curve for the 100 year period.
2.29mins
Q7a

Small countries trying to develop an industrial economy rapidly often try to achieve their objectives by importing foreign capital and technology. Statistics Canada data show that when Canada attempted this strategy from 1867 to 1967, the amount of U.S. investment in Canada increased from about represented by the simple mathematical model C(t) = 0.015 \times 10^9 e^{0.07533t}, where t represents the number of years (starting with 1967 as zero) and C represents the total capital investment from US sources in dollars.

• Compare the growth rate of US investment in 1947 with the rate in 1967.
1.35mins
Q7b

Small countries trying to develop an industrial economy rapidly often try to achieve their objectives by importing foreign capital and technology. Statistics Canada data show that when Canada attempted this strategy from 1867 to 1967, the amount of U.S. investment in Canada increased from about represented by the simple mathematical model C(t) = 0.015 \times 10^9 e^{0.07533t}, where t represents the number of years (starting with 1967 as zero) and C represents the total capital investment from US sources in dollars.

• Determine the growth rate of investment in 1967 as a percent of the amount invested.
0.28mins
Q7c

A colony of bacteria in a culture grows at a rate given by N(t) = 2^{\frac{t}{5}}, where N is the number of bacteria t minutes form the beginning. The colony is allowed to grow fro 60 min, at which time a drug is introduced to kill the bacteria. The number of bacteria killed is given by K(t) = e^{\frac{t}{3}}, where K bacteria are killed at time t minutes.

a. Determine the maximum number of bacteria present and the time at which this occurs.

b. Determine the time at which the bacteria colony is obliterated.

5.55mins
Q8

Lorianne is studying for two different exams. Because of the nature of the courses, the measure of study effectiveness on a scale from 0 to 10 for the first course is

E_1 = 0.6(9 + te^{-\frac{t}{20}}), while the measure for the second course

is E_2 = 0.5(10 + te^{-\frac{t}{10}}) . Lorianne is prepared to spend up to 30 h, in total, studying for the exams. The total effectiveness is given by f(t) = E_1 + E_2.

How should this time be allocated to maximize total effectiveness?

Q9

Find the absolute extrema of f(x) =x = e^{2x} on the interval x\in [-2, 2].

1.59mins
Q10

For f(x) = x^2e^x, determine the intervals of increase and decrease.

1.31mins
Q11a

Determine the absolute minimum value of f(x) = x^2e^x.

0.30mins
Q11b

Find the maximum and minimum values.

y = e^x + 2

0.17mins
Q12a

Find the maximum and minimum values.

y = xe^x + 3

1.25mins
Q12b

Find the maximum and minimum values. Sketch the function.

y = 2xe^{2x}

1.15mins
Q12c

Find the maximum and minimum values. Sketch the function.

y = 3xe^{-x} + x

1.25mins
Q12d

The profit function of a commodity is P(x) = xe^{-0.5x^2}, where x > 0. Find the maximum value of the function, if x is measured in hundreds of units and P, is measured in thousands of dollars.

Q13

a. Determine an equation for  A(t)  , the amount of money in the account at any time  t  .

b. Find the derivative  A^{\prime}(t)  of the function.

c. At what rate is the amount growing at the end of two years? At what rate is it growing at the end of five years and at the end of 10 years?

d. Is the rate constant?

e. Determine the ratio of  \frac{A^{\prime}(t)}{A(t)}  for each value that you determined for  A^{\prime}(t)  .

f. What do you notice?